Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients

Abstract

In a recent paper Bray and Pinsky [1] estimated the growth of f ̂ ( ξ ) , the Fourier transform of f ( x ) where x ∈ R d , by some moduli of smoothness. We show here that noticeably better results can be derived as an immediate corollary of previous theorems in [2]. The improvements include dealing with higher levels of smoothness and using the fact that for higher dimensions (when d ≥ 2 ) the description of smoothness requires less information. Using a similar technique, we also deduce relations between the smoothness of f ( x ) for x ∈ S d − 1 or x ∈ T d and the growth of the coefficients of the expansion by spherical harmonic polynomials or trigonometric polynomials.